Band centrifugation of macromolecules in self-generating density gradients. III. Conditions for convection-free band sedimentation

The conditions for convection-free hand sedimentation are analyzed in terms of the negative density gradients associated with the leading edge of a band and the positive density gradients generated during the experiment. The amount of material necessary to perform a band-centrifugation experiment depends on the diffusion coefficient of the macromolecules, which determines the rate at which the concentration at band maximum decreases, and on the extinction coefficient at the wavelength of observation. The maximum negative gradients in bands of macromolecules with diffusion and extinction coefficients typical of proteins, nucleic acids, and viruses are calculated. Positive density gradients generated by diffusion of small molecules between the thin lamella and the bulk solution are calculated for bulk solutions such us 1M NaCl or 95% D₂O. These “diffusion gradients” are generally adequate to stabilize bands of the above macromolecules. Positive density gradients generated by sedimentation of salts within the bulk solution may be significant in providing stability near the bottom of the cell. The effects of inadequate stabilizing gradients are discussed, and are found to cause forward spreading of the band.     

Macromolecular size-and-shape distributions by sedimentation velocity analytical ultracentrifugation

Sedimentation velocity analytical ultracentrifugation is an important tool in the characterization of macromolecules and nanoparticles in solution. The sedimentation coefficient distribution c(s) of Lamm equation solutions is based on the approximation of a single, weight-average frictional coefficient of all particles, determined from the experimental data, which scales the diffusion coefficient to the sedimentation coefficient consistent with the traditional s approximately M(2/3) power law. It provides a high hydrodynamic resolution, where diffusional broadening of the sedimentation boundaries is deconvoluted from the sedimentation coefficient distribution. The approximation of a single weight-average frictional ratio is favored by several experimental factors, and usually gives good results for chemically not too dissimilar macromolecules, such as mixtures of folded proteins. In this communication, we examine an extension to a two-dimensional distribution of sedimentation coefficient and frictional ratio, c(s,f(r)), which is representative of a more general set of size-and-shape distributions, including mass-Stokes radius distributions, c(M,R(S)), and sedimentation coefficient-molar mass distributions c(s,M). We show that this can be used to determine average molar masses of macromolecules and characterize macromolecular distributions, without the approximation of any scaling relationship between hydrodynamic and thermodynamic parameters.     

Size-distribution analysis of macromolecules by sedimentation velocity ultracentrifugation and lamm equation modeling

A new method for the size-distribution analysis of polymers by sedimentation velocity analytical ultracentrifugation is described. It exploits the ability of Lamm equation modeling to discriminate between the spreading of the sedimentation boundary arising from sample heterogeneity and from diffusion. Finite element solutions of the Lamm equation for a large number of discrete noninteracting species are combined with maximum entropy regularization to represent a continuous size-distribution. As in the program CONTIN, the parameter governing the regularization constraint is adjusted by variance analysis to a predefined confidence level. Estimates of the partial specific volume and the frictional ratio of the macromolecules are used to calculate the diffusion coefficients, resulting in relatively high-resolution sedimentation coefficient distributions c(s) or molar mass distributions c(M). It can be applied to interference optical data that exhibit systematic noise components, and it does not require solution or solvent plateaus to be established. More details on the size-distribution can be obtained than from van Holde-Weischet analysis. The sensitivity to the values of the regularization parameter and to the shape parameters is explored with the help of simulated sedimentation data of discrete and continuous model size distributions, and by applications to experimental data of continuous and discrete protein mixtures.     

A model for sedimentation in inhomogeneous media. II. Compressibility of aqueous and organic solvents

The effects of solvent compressibility on the sedimentation behavior of macromolecules as observed in analytical ultracentrifugation are examined. Expressions for the density and pressure distributions in the solution column are derived and combined with the finite element solution of the Lamm equation in inhomogeneous media to predict the macromolecular concentration distributions under different conditions. Independently, analytical expressions are derived for the sedimentation of non-diffusing particles in the limit of low compressibility. Both models are quantitatively consistent and predict solvent compressibility to result in a reduction of the sedimentation rate along the solution column and a continuous accumulation of solutes in the plateau region. For both organic and aqueous solvents, the calculated deviations from the sedimentation in incompressible media can be very large and substantially above the measurement error. Assuming conventional configurations used for sedimentation velocity experiments in analytical ultracentrifugation, neglect of the compressibility of water leads to systematic errors underestimating sedimentation coefficients by approximately 1% at a rotor speeds of 45000 rpm, but increasing to 2-5% with increasing rotor speeds and decreasing macromolecular size. The proposed finite element solution of the Lamm equation can be used to take solvent compressibility quantitatively into account in direct boundary models for discrete species, sedimentation coefficient distributions or molar mass distributions. Using the analytical expressions for the sedimentation of non-diffusing particles, the ls-g*(s) distribution of apparent sedimentation coefficients is extended to the analysis of sedimentation in compressible solvents. The consideration of solvent compressibility is highly relevant not only when using organic solvents, but also in aqueous solvents when precise sedimentation coefficients are needed, for example, for hydrodynamic modeling.     

Determination of sedimentation coefficients for small peptides.

Direct fitting of sedimentation velocity data with numerical solutions of the Lamm equations has been exploited to obtain sedimentation coefficients for single solutes under conditions where solvent and solution plateaus are either not available or are transient. The calculated evolution was initialized with the first experimental scan and nonlinear regression was employed to obtain best-fit values for the sedimentation and diffusion coefficients. General properties of the Lamm equations as data analysis tools were examined. This method was applied to study a set of small peptides containing amphipathic heptad repeats with the general structure Ac-YS-(AKEAAKE)nGAR-NH2, n = 2, 3, or 4. Sedimentation velocity analysis indicated single sedimenting species with sedimentation coefficients (s(20,w) values) of 0.37, 0.45, and 0.52 S, respectively, in good agreement with sedimentation coefficients predicted by hydrodynamic theory. The described approach can be applied to synthetic boundary and conventional loading experiments, and can be extended to analyze sedimentation data for both large and small macromolecules in order to define shape, heterogeneity, and state of association.     

Anomalies in sedimentation. 3. A model for the inherent instability of solutions of very large particles in high centrifugal fields

Sedimentation, diffusion, and hydrodynamic interaction are included in a simplified model for the behavior of a solution macromolecules in a centrifugal field. The model is simulated by a program on a digital computer. As the diffusion coefficient is allowed to decrease, the simulation shows that the molecules tend to cluster in high centrifugal fields. The clusters move through the solution in the direction of sedimentation in a wave-like fashion, acquiring new molecules at the front and shedding old ones to the rear. It is suggested that this tendency to cluster could account for the loss of DNA and other very large molecules from solution which is observed to occur at high centrifugal fields.     

A method for measuring sedimentation and diffusion of macromolecules in capillary tubes by total intensity and quasi-elastic light-scattering techniques.

Light scattered from a macromolecular solution in a capillary tube is used to determine both the sedimentation and translational diffusion coefficients. The capillary tube is spun in a preparative centrifuge, removed, and placed in a light-scattering photometer equipped with a scanning mechanism. The intensity distribution of scattered light along the tube represents the concentration profile in the tube and provides the measure of boundary migration. The sedimentation coefficient is determined from this measure and the applied centrifugal field. The diffusion coefficient is obtained from a time-autocorrelation analysis of fluctuations in intensity of light scattered from any fixed point of the profile. These coefficients were obtained for two monodisperse systems, R17 bacteriophage and 28s ribosomal rat liver RNA. The molecular weights obtained from ratios of these coefficients are in good agreement with literature values. In the sedimentation analysis, deviations from linearity between boundary displacement and applied field were found to be less than 1%. This precision confirms that the boundary is stable for the capillary geometry even in the absence of a preformed density gradient. The sedimentation coefficients of identical samples were also measured with the Spinco Model E analytical ultracentrifuge; results of the two methods agree to within 4%. As a consequence of the capillary tube geometry and light-scattering detection, sedimentation coefficients can be obtained from sample volumes of less than 100 μl. This detection techniques is thus far demonstrated to be at least an order of magnitude more sensitive than Schlieren optics, thereby useful when uv absorption is not applicable. For diffusion measurements there are also several inherent advantages. The diffusion coefficient is obtained from the identical sample, and scanning provides the capability to measure D from various parts of the sedimentation profiles and thereby directly explore concentration dependence, homogeneity, and integrity of the sample. The capillary tube with a layer of silicone oil over the sample and centrifugation provides an effective method to cleanse the solution and trap all dust.     

An improved approximate solution of the Lamm equation for the simultaneous estimation of sedimentation and diffusion coefficients from sedimentation velocity experiments

Sedimentation and diffusion coefficients are important parameters to describe size and shape of macromolecules in solution. The data can be obtained from sedimentation velocity experiments by a nonlinear fitting procedure using approximate solutions for the Lamm equation. Here, we present a modification of such a model function that was originally proposed by Fujita [H. Fujita, Mathematical Theory of Sedimentation Analysis, Wiley, New York, 1962]. The extended model function is well suitable to study low molecular mass compounds. The improvement of this solution given here is based on using an adjustable value for the explicit integration variable, z, the reduced radius. This modification leads to more accurate sedimentation and diffusion coefficients compared to using a constant value of 0.5 as used by Fujita. The advantage of our modification was demonstrated by the analysis of noise-free curves calculated using the finite element method, as well as experimental curves obtained for the peptides angiotensin I and II. The relatively low sedimentation and diffusion coefficients found for both substances indicate that the peptides exist as extended chains of about 3.65 nm (angiotensin I) or 3.04 nm length (angiotensin II) in solution. The lack of higher-order structure of the peptides that was derived also from CD spectra might facilitate receptor binding, and could be one reason for the fast proteolytic digestion of the free peptides.     

Band centrifugation of macromolecules in self-generating density gradients. II. Sedimentation and diffusion of macromolecules in bands

Three approaches to the simultaneous sedimentation and diffusion of hands or zones of noninteracting homogeneous macromolecules are examined: (1) The authors' method of moments: (2) the transport me of Sehumaker and Rosenbloom; and (3) the stochastic solution of the Lamm equation due to Gehatia and Katehalski. All three methods indicate that the motion of the maximum of the hand may be used to evaluate the sedimentation coefficient. The moment, method provides relations which appear to be useful for measuring diffusion coefficients. Relations are given for the analysis of resolved components. The problem of measuring sedimentation coefficients of macromolecules with concentration-dependent sedimentation coefficients is examined. Methods are described for evaluating the sedimentation coefficient in these systems and for obtaining the sedimentation coefficient at infinite dilution. Methods are described for determining the weight-average sedimentation coefficient in Multi-component systems, and the differential and integral distribution of sedimentation coefficients of macromolecules with low-diffusion coefficients.     

A model for water motion in crystals of lysozyme based on an incoherent quasielastic neutron-scattering study

This paper reports an incoherent quasielastic neutron scattering study of the single particle, diffusive motions of water molecules surrounding a globular protein, the hen egg-white lysozyme. For the first time such an analysis has been done on protein crystals. It can thus be directly related and compared with a recent structural study of the same sample. The measurement temperature ranged from 100 to 300 K, but focus was on the room temperature analysis. The very good agreement between the structural and dynamical studies suggested a model for the dynamics of water in triclinic crystals of lysozyme in the time range approximately 330 ps and at 300 K. Herein, the dynamics of all water molecules is affected by the presence of the protein, and the water molecules can be divided into two populations. The first mainly corresponds to the first hydration shell, in which water molecules reorient themselves fivefold to 10-fold slower than in bulk solvent, and diffuse by jumps from hydration site to hydration site. The long-range diffusion coefficient is five to sixfold less than for bulk solvent. The second group corresponds to water molecules further away from the surface of the protein, in a second incomplete hydration layer, confined between hydrated macromolecules. Within the time scale probed they undergo a translational diffusion with a self-diffusion coefficient reduced approximately 50-fold compared with bulk solvent. As protein crystals have a highly crowded arrangement close to the packing of macromolecules in cells, our conclusion can be discussed with respect to solvent behavior in intracellular media: as the mobility is highest next to the surface, it suggests that under some crowding conditions, a two-dimensional motion for the transport of metabolites can be dominant.     

A new approximate whole boundary solution of the Lamm differential equation for the analysis of sedimentation velocity experiments

Sedimentation velocity is one of the best-suited physical methods for determining the size and shape of macromolecular substances or their complexes in the range from 1 to several thousand kDa. The moving boundary in sedimentation velocity runs can be described by the Lamm differential equation. Fitting of suitable model functions or solutions of the Lamm equation to the moving boundary is used to obtain directly sedimentation and diffusion coefficients, thus allowing quick determination of size, shape and other parameters of macromolecules. Here we present a new approximate whole boundary solution of the Lamm equation that simultaneously allows the specification of sedimentation and diffusion coefficients with deviations smaller than 1% from the expected values.     

Conformation parameters of linear macromolecules from velocity sedimentation and other hydrodynamic methods

Linear macromolecules constitute a broad class of synthetic and natural polymers which are highly useful in various technologies and represent the key molecular systems in living nature. The study of the molecular characteristics of these polymers represents an important problem in fundamental and applied science. The methods of molecular hydrodynamics have been and remain an important way of studying the molar mass, molar mass distribution, size and conformation of linear polymers. This paper discusses the approaches to the problems of hydrodynamic methods, in particular analytical velocity ultracentrifugation, in the study of various types of linear macromolecule. The velocity sedimentation data were processed with three different methods: Sedanal and Sedfit software, and the classical approach of evaluating the rate at which the sedimentation boundary moves. The Sedfit program also allows an evaluation of the frictional ratio values, i.e., the coefficient of translational diffusion. It will be discussed for which systems the estimation of the frictional ratio obtained by Sedfit is adequate and for which it is not. The applications of other hydrodynamic methods (intrinsic viscosity, translational diffusion) are also discussed with a view to obtaining the conformational characteristics of linear macromolecules.     

Fluorescence correlation spectroscopy and nonideal solutions

The information that may be obtained from a fluorescence correlation spectroscopic study of a nonideal solution is considered. If all of the macromolecules in a two-component solution are fluorescently labeled, the mutual diffusion coefficient will be measured. If only a few of the macromolecules in a solution are fluorescently labeled, the tracer diffusion coefficient will be obtained. Two nonideal systems that probably may usefully be studied with fluorescence correlation spectroscopy are proposed. The application of fluorescence correlation spectroscopy to studies of lateral diffusion in biological membranes is discussed; the form of the contribution to the fluorescence correlation spectrum of bulk motion within a membrane is noted.     

A parameterized overspeeding method for the rapid attainment of low-speed sedimentation equilibrium

The approach of a solution of dilute, monodisperse, globular macromolecules to low-speed sedimentation equilibrium in an ultracentrifuge is simulated by numerical integration of the Lamm equation. Various combinations of overspeed time and angular velocity are used to assess the conditions needed to minimize the time it takes the solution to attain sedimentation equilibrium. The optimal overspeeding time and angular velocity are determined over a wide range of values of the molecular weight (relative molar mass) of the solute and the radial distance between the meniscus and base of the solution. The results may be expressed as simple functions which allow facile calculation of (a) the optimal overspeeding time and velocity, and (b) the time required to reach sedimentation equilibrium. The results are in reasonable agreement with previous analytical solutions which were based on several simplifying assumptions. The parameterized overspeeding procedure is shown to be robust over a wide range of conditions, and typically leads to a greater than 5-fold reduction in centrifugation time.     

Hydrodynamic properties of human cervical-mucus glycoproteins in 6M-guanidinium chloride.

Cervical mucins and fragments thereof were studied by sedimentation-velocity, rotatory viscometry and laser light-scattering performed as photon-correlation spectroscopy as well as low-angle total-intensity measurements. The Mr of the whole mucins is 10 X 10(6)-15 X 10(6), whereas fragments obtained after reduction of disulphide bonds ('subunits') have Mr 2.1 X 10(6)-2.9 X 10(6), depending on the method used. Subsequent trypsin digestion of subunits afforded glycopeptides with Mr approx. 0.4 X 10(6). The high frictional ratio for the whole mucins is interpreted as a large degree of expansion. The Stokes radius calculated from the diffusion coefficient is approx. 110nm for the whole mucins, which is in agreement with that estimated from the radius of gyration (130nm) by using the concept of the equivalent hydrodynamic sphere. The ratio of the concentration-dependence parameter for the reciprocal sedimentation coefficient (Ks) to the intrinsic viscosity ( [eta] ) for the whole mucins is 1.42, suggesting that the individual macromolecule occupies a spheroidal domain in solution. The relationship between [eta] and Mr for whole mucins, subunits and T-domains suggests that they are linear flexible macromolecules behaving as somewhat 'stiff' random coils. This conclusion is supported by the relationships between the sedimentation coefficients, the diffusion coefficients and the Mr. The hydrodynamic behaviour of the mucins is thus close to that expected for coiling macromolecules entrapping a lot of solvent, which is consistent with the postulated polymeric structure.     

Analysis of the ligand-promoted global conformational change in aspartate transcarbamoylase. Evidence for a two-state transition from boundary spreading in sedimentation velocity experiments

A global conformational change in the regulatory enzyme aspartate transcarbamoylase of Escherichia coli was demonstrated 20 years ago by the 3.5% decrease in the sedimentation coefficient of the enzyme upon its interaction with carbamoyl phosphate and saturating amounts of the aspartate analog succinate. This "swelling" of aspartate transcarbamoylase attributable to the T----R allosteric transition was observed also in subsequent studies when the enzyme was completely saturated with the bisubstrate analog N-(phosphonacetyl)-L-aspartate. In neither of these studies was a direct attempt made by an analysis of boundary spreading (expressed as an apparent diffusion coefficient) on partially liganded enzyme to determine whether the solution contained only T and R-state molecules, as expected for a concerted transition, or a mixture of more than two distinct conformational states. The sensitivity of boundary spreading measurements was tested with a known mixture of fully liganded wild-type enzyme (R-state) and an inactive T-state mutant that did not bind either succinate or the bisubstrate ligand. This experiment yielded broad boundaries with an apparent diffusion coefficient about 10% greater than that of T-state enzyme, due to the differential sedimentation of the two independent species. Identical boundary spreading was obtained theoretically by simulating an equimolar mixture of T and R-state aspartate transcarbamoylase. These results proved that the boundary spreading measurement was sensitive to the presence of heterogeneity. Analogous experiments with only wild-type enzyme in the presence of sub-stoichiometric amounts of the tightly bound bisubstrate ligand sufficient to promote a 1.8% decrease in sedimentation coefficient also exhibited broader boundaries, corresponding to a 10% increase in the apparent diffusion coefficient relative to the unliganded enzyme. In contrast, such broad boundaries were not observed in experiments when the weakly bound succinate was present in quantities sufficient to cause the same 1.8% decrease in sedimentation coefficient. The differences in boundary spreading observed with the two active-site ligands were accounted for by the affinities of the respective ligands for the enzyme and the transport theory of a ligand-promoted isomerization of the protein. In the presence of sub-stoichiometric levels of the tight-binding bisubstrate ligand, the dynamic equilibrium between the T and the R-state is essentially uncoupled and the species sediment at slightly different rates to give broad boundaries.(ABSTRACT TRUNCATED AT 250 WORDS)     

Enhanced resolution of sedimentation coefficient distribution profiles by extrapolation to infinite time

Sedimentation velocity experiments can be used to identify two or more independent non-interacting macromolecules, which differ in their size by only a few percent. The procedure requires the extrapolation of differential apparent sedimentation coefficient distributions obtained at different running time to t → ∞ and works because it eliminates or greatly reduces diffusion effects. Here, we present an improved time extrapolation function of sedimentation distribution profiles originally presented by Stafford (In: Harding, Rowe, Horton (eds.) Analytical ultracentrifugation in biochemistry and polymer science, 1992). We describe a computing procedure with the program lamm to analyze concentration profiles obtained by absorbance or interference optics that utilizes suitable smoothing methods for noisy data sets and present examples which include time invariant noises.     

Kinetics of sedimentation in a density gradient

Two models of sedimentation in a density gradient are analyzed. The first is for sedimentation in cylindrical sector geometry and contains the assumption that diffusion can be neglected. The second treats sedimentation in a rectangular field and includes diffusion, although the boundaries are not treated exactly. In both of these models we approximate the time dependence of the gradient by a relaxation form. We derive exact results for both models. It is also shown that the sedimentation coefficient can be calculated from data by following the motion of the position of the maximum (or minimum) of the concentration gradient.     

The concentration-dependence of macromolecular parameters.

Theories concerning the concentration-dependence of sedimentation and diffusion coefficients for macro-molecules in dilute solution are compared and discussed, together with their experimental basis. An attempt has been made to clarify an important uncertainty still present in the literature as to whether sedimentation coefficients should be corrected for solvent or solution density. It is pointed out that the two processes yield the same extrapolation limit but different concentration-dependencies, which have, however, been related. A general expression is derived thermodynamically for the concentration-dependence of diffusion that includes the coefficient of the concentration term involved in sedimentation (on the basis of sedimentation coefficients corrected from solution density). For rigid spherical particles the expression is shown to be exactly equivalent to one given by Batchelor [(1976) J. Fluid Mech. 74, 1-29], which was derived on the basis of sedimentation coefficients corrected from solvent density. Finally, we discuss the concentration-dependence of apparent weight-average relative molecular masses ('molecular weights') (from, e.g., sedimentation equilibrium) and note an important omission in some earlier representations.     

Quasi-elastic light scattering of Artemia ribosomes sedimented in a CsCl density gradient

It is impossible to measure the diffusion coefficient of macromolecules directly and accurately by quasi—elastic light scattering, when aggregates cannot be eliminated from the solutions to be investigated. Nevertheless, a simple method can be applied to overcome this problem in many cases. Aggregates are separated from the monomeric macromolecules by rate-zonal sedimentation in a CsCl density gradient in a transparent centrifugation tube; the monomers are then located by laser light scattering intensity measurements; photon correlation spectroscopy of the scattered light finally yields their diffusion coefficient. The viscosity of aqueous CsCl solutions at different temperatures and concentrations allows a good separation by centrifugation and a low uncertainty in the reduction of the measured diffusion coefficient to standard conditions.The application of the method to eukaryotic large ribosomal subunits is described as an example.